Convergence of probabilities implies convergence of char. functions

Question

let $X_n, X$ are discrete random variables. Does $P(X_n = k) \rightarrow P(X=k)$, as $n \rightarrow +\infty$ for all $k \in N$ implies convergence of corresponding characteristic functions: $\varphi_{X_n}(t) \rightarrow \varphi_X(t)$ as $n\to +\infty$ ?

Thanks

Answer 1

Yes.

1. Fix $\epsilon$ and choose $M$ so large that $\sum_{k=1}^M P(X=k) > 1-\epsilon$.

2. Show that there exists $N$ so large that $\sum_{k=1}^M P(X_n = k) > 1-2\epsilon$ for all $n \ge N$.

3. Write $E[e^{itX_n}] = \sum_{k=1}^M e^{itk} P(X_n = k) + \sum_{k=M+1}^\infty e^{itk} P(X_n=k)$. Note that the first term behaves well and the second term is bounded in absolute value by $2\epsilon$ for all sufficiently large $N$.

4. Conclude $\limsup_{n \to \infty} \left|E[e^{itX_n}] - E[e^{itX}]\right| \le 3 \epsilon$, and $\epsilon$ was arbitrary.